Lindel¨ofRepresentations and (Non-)Holonomic Sequences Philippe Flajolet Algorithms Project, INRIA Rocquencourt, F-78153 Le Chesnay (France) Philippe.Flajolet AT inria.fr Stefan Gerhold∗ TU Vienna (Austria) and Microsoft Research-INRIA, Orsay (France) sgerhold AT fam.tuwien.ac.at Bruno Salvy Algorithms Project, INRIA Rocquencourt, F-78153 Le Chesnay (France) Bruno.Salvy AT inria.fr Submitted: Jun 9, 2009; Accepted: Dec 9, 2009; Published: Jan 5, 2010 Mathematics Subject Classifications: 11B83, 30E20, 33E20 Keywords: Holonomic sequence, D-finite function, Lindel¨ofrepresentation Abstract Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindel¨of,which belong to an attractive but somewhat neglected chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences, and, accordingly, the non-existence of lin- ear differential equations with polynomial coefficients annihilating their generating functions. In particular, the corresponding generating functions are transcendental. Asymptotic estimates of certain finite difference sequences come out as a byproduct of the Lindel¨ofapproach. Introduction There has been recently a surge of interest in methods for proving that certain sequences coming from analysis or combinatorics are non-holonomic. Recall that a sequence (fn) ∗This work was partially supported by CDG, BA-CA, AFFA, and the joint INRIA-Microsoft Research Laboratory. the electronic journal of combinatorics 17 (2010), #R3 1 is holonomic, or P -recursive, if it satisfies a linear recurrence with coefficients that are polynomial (equivalently, rational) in the index n; that is, d X pk(n)fn−k = 0, pk(n) ∈ C[n], p0 6≡ 0. k=0 Put otherwise, its generating function f(z), called holonomic or D-finite, satisfies a linear differential equation with coefficients that are polynomial (equivalently, rational) in the P n variable z; that is, with f(z) := fnz , n>0 e X dk q (z) f(z) = 0, q (z) ∈ [z], q 6≡ 0. k dzk k C e k=0 Within combinatorics, the holonomic framework has been largely developed by Stanley, Zeilberger, Lipshitz, and Gessel [18, 25, 26, 35, 44], who provided a rich set of closure properties satisfied by the holonomic class. Since the class of holonomic functions contains all algebraic functions, establishing that a sequence is non-holonomic can in particular be regarded as a strong transcendence result for its generating function. In recent years, proofs have appeared of the non-holonomic character of sequences such as √ 1 log n, n, nn, , $ , log log n, ζ(n), H n √ n √ n!, arctan(n), n2 + 1, ee1/n where Hn is a harmonic number, $n represents the nth prime number, and ζ(s) is the Riemann zeta function. The known proofs range from elementary [16] to algebraic and analytic [2, 10, 22, 23]. The present paper belongs to the category of complex-analytic approaches and it is, to a large extent, a sequel to the paper [10]. Keeping in mind that a univariate holonomic function can have only finitely many singularities, we enunciate the following general principle. Holonomicity criterion. The shape of the asymptotic expansion of a holonomic function at a singularity z0 is strongly constrained, as it can only involve, in sectors of C, (finite) linear combinations of “elements” of the form ∞ −1/r α X js exp P (Z ) Z Qj(log Z)Z ,Z := (z − z0), (1) j=0 with P a polynomial, r an integer, α a complex number, s a rational of Q>0 and the Qj a family of polynomials of uniformly bounded degree. (For an expansion at infinity, change Z to Z := 1/z.) Therefore, any function whose asymptotic structure at a singularity (possibly infinity) is incompatible with elements of the form (1) must be non-holonomic. the electronic journal of combinatorics 17 (2010), #R3 2 Equation (1) is a paraphrase of the classical structure theorem for solutions of linear differential equations with meromorphic coefficients [20, 39]; see also Theorem 2 of [10] and the surrounding comments. What the three of us did in [10] amounts to implementing the principle above, in combination with a basic Abelian theorem. Functional expansions departing from (1) can then be generated, by means of such a theorem, from corresponding terms in the asymptotic expansions of sequences: for instance, quantities such as 1 √ log log n, , plog n, e log n,... log n constitute forbidden “elements” in expansions of holonomic sequences—hence their pres- ence immediately betrays a non-holonomic√ sequence. In proving the non-holonomic char- acter of the sequences (log n) and ( n), we could then simply observe in [10] that the nth order differences (this is a holonomicity-preserving transformation) involve log log n and 1/ log n in an essential way. What we do now, is to push the method further, but in another direction, namely, that of Lindel¨ofrepresentations of generating functions. Namely, for a suitable “coefficient function” φ(s), one has ∞ X 1 Z 1/2+i∞ π φ(n)(−z)n = − φ(s)zs ds, (2) 2iπ sin πs n=1 1/2−i∞ where the left side is, up to an alternating sign, the generating function of the se- quence (φ(n)). Based on representations of type (2), we determine directly the asymp- totic behaviour at infinity of the generating functions of several sequences given in closed form, and detect cases that contradict (1), hence entail the non-holonomicity of the se- quence (φ(n)). Here are typical results that can be obtained by the methods we develop. θ Theorem 1. (i) The sequences ecn , with c, θ ∈ R, are non-holonomic, except in the trivial cases c = 0 or θ ∈ {0, 1}. In particular, √ √ e n, e− n, e1/n, e−1/n (3) are non-holonomic. (ii) The sequences √ 1 1 √ Γ(n 2) 1 , , Γ(n 2), √ , Γ(ni), (4) 2n ± 1 n! + 1 Γ(n 3) ζ(n + 2) are also non-holonomic. The non-holonomicity of the sequences in (3) also appears in the article by Bell et al. [2], where it is deduced from an elegant argument involving Carlson’s Theorem com- bined with the observation that φ(s) = ecsθ is non-analytic and non-polar at 0. The the electronic journal of combinatorics 17 (2010), #R3 3 results of [2] do usually not, however, give access to the cases where the function φ(s) is either meromorphic throughout C or entire. In particular, they do not seem to yield the non-holonomic character of the sequences listed in (4), except for the first one. (In- deed, 1/(2n ± 1) has been dealt with in [2] by an extension of the basic method of that paper. Moreover, any potential holonomic recurrence for 1/(2n ± 1) can be refuted by an elementary limit argument, communicated by Fr´ed´eric Chyzak and transcribed in [17, Proposition 1.2.2].) A great advantage of the Lindel¨of approach is that it leads to precise asymptotic expansions that are of independent interest. For instance, in §2.2 below we will encounter the expansion (−z)n π 1 X X sk ∼ − 0 z , z → ∞, n! + 1 sin πsk Γ (sk + 1) n>1 k>1 where (s ) ≈ (−3.457, −3.747, −5.039, −5.991,... ) k k>1 are the solutions of Γ(s+1) = −1. This formula features a remarkable structural difference to the superficially resembling function P (−z)n/n! = exp(−z) − 1. n>1 Plan of the paper. The general context of Lindel¨of representations is introduced in Section 1; in particular, Theorem 2 of that section provides detailed conditions granting us the validity of (2). As we show next, these representations make it possible to analyse the behaviour of generating functions towards +∞, knowing growth and singularity properties of the coefficient function φ(s) in the complex plane. The global picture is a general correspondence of the form: location s0 of singularity of φ(s) −→ singular exponent of F (z) nature of singularity of φ(s) −→ logarithmic singular elements in F (z). (This is, in a way, a dual situation to singularity analysis [11, 13].) Precisely, three major cases are studied here; see Figure 2 below for a telegraphic summary. — Polar singularities. When φ(s) can be extended into a meromorphic function, the asymptotic expansion of the generating function of the sequence (φ(n)) at infinity can be read off from the poles of φ(s). Section 2 gives a detailed account of the corresponding “dictionary”, which is in line with early studies by Ford [15].√ It implies the non-holonomicity of sequences such as 1/(2n − 1), 1/(n! + 1), Γ(n 2). — Algebraic singularities. In this case, a singularity of exponent −λ in the function φ(s) essentially induces a term of the form (log z)λ−1 in the generating function as z → +∞. We show the phenomena at stake by performing√ a detailed asymptotic study of the generating functions of sequences such as e n in Section 3, based on the use of Hankel contours. The non-holonomic character of the sequences e±nθ for θ ∈ ]0, 1[ arises as a consequence. the electronic journal of combinatorics 17 (2010), #R3 4 Coefficients φ(n) z → ∞ z → −1 √ e2 log z 1 e1/n − √ 2 π(log z)1/4 1 + z √ 1 e−1/n − √1 (log z)−1/4 cos 2 log z − 1 π π 4 1 + z √ √ 1 πe−1/8 1 e n −1 − √ exp π log z (1 + z)3/2 4(1 + z) √ 1 e− n −1 + √ E(1) + E0(1)(1 + z) π log z Figure 1: Asymptotic forms of E(z; c, θ), for representative parameter values.